Wavelet Projection Research Articles

Machine learning-based vorticity evolution and super-resolution of homogeneous isotropic turbulence using wavelet projection

A wavelet-based machine learning method is proposed for predicting the time evolution of homogeneous isotropic turbulence where vortex tubes are preserved. Three-dimensional convolutional neural networks and long short-term memory are trained with a time series of direct numerical simulation (DNS) data of homogeneous isotropic turbulence at the Taylor microscale Reynolds number 92. The predicted results are assessed by using the flow visualization of vorticity and statistics, e.g., probability density functions of vorticity and enstrophy spectra. It is found that the predicted results are in good agreement with DNS results. The small-scale flow topology considering the second and the third invariants of the velocity gradient tensor likewise shows an approximate match. Furthermore, we apply the pre-trained neural networks to coarse-grained vorticity data using super-resolution. It is shown that the super-resolved flow field well agrees with the reference DNS field, and thus small-scale information and vortex tubes are well regenerated.

A Novel Image Segmentation Technique with N-L Means Filter Using Region Merging Strategy

Image segmentation is a branch of digital image processing which emphases on splitting an image into different fragments according to their features. This work proposes a novel image segmentation technique using fast non local (N-L) means filter with region merging approach. Here, a random noise is added to the original image and discrete wavelet transform (DWT) is applied to separate the low and high frequency bands of the digital image. The low frequency component is subjected to a fast N-L means filter and the filtered image is segmented by graph-based segmentation technique. Here, these segmented image regions are merged to reduce the over segmentation patterns. The detailed coefficients of the image component are treated with soft threshold filter to reduce the residual noise. Finally merged image is combined with preserved edge features at wavelet projection resulting in better segmented image. The results are compared with some of the best methods available in literature and are found to be significantly better. Keywords: Discrete wavelet transform (DWT), fast NL-means filter, graph-based segmentation, boundary based hierarchical region merging, soft-thresholding, peak signal to noise ratio (PSNR).

Wavelet Estimation of Function Derivatives from a Multichannel Deconvolution Model

This paper considers a multichannel deconvolution model with Gaussian white noises. The goal is to estimate the d -th derivatives of an unknown function in the model. For super-smooth case, we construct an adaptive linear wavelet estimator by wavelet projection method. For regular-smooth case, we provide an adaptive nonlinear wavelet estimator by hard-thresholded method. In order to measure the global performances of our estimators, we show upper bounds on convergence rates using the L p -risk ( 1 ≤ p < ∞ ).

The Convergence of Operator With Rapidly Decreasing Wavelet Functions

The expansion of (2D) wavelet functions with respect to Lp(R2) space converging almost everywhere for 1<p<∞ throughout the length of the Lebesgue set points of space functions is investigated in this research. The convergence is established by assuming some wavelet function minimal regularity ψj1,j2,k1,k2 under the current description of the wavelet projection operator known as 2D Hard Sampling Operator. Note that the feature of fast decline in 2D is derived here. Another condition is used, for instance, the wavelet expansion's boundedness under the Hard Sampling Operator. The bound (limit) is governed in magnitude with respect to the maximal equality of the Hardy-Littlewood maximal operator. Some ideas presented in this work are to find a new method to prove the convergence theory for a new type of conditional wavelet operator. Propose some conditions for wavelets functions and their expansion can support the operator to be convergence. It also performs a comparison with the identity convergent operator is our method for achieving this convergence.

Pointwise Wavelet Estimations for a Regression Model in Local Hölder Space

This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in local Hölder space. A nonlinear wavelet estimator is provided by the hard thresholding method in order to obtain an adaptive estimator. The convergence rate of the nonlinear estimator is the same as the linear estimator up to a logarithmic term. Finally, it should be pointed out that the convergence rates of two wavelet estimators are consistent with the optimal convergence rate on pointwise nonparametric estimation.

Wavelet representation of singular integral operators

This article develops a novel approach to the representation of singular integral operators of Calderón–Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderón–Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of T(1) theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the \(A_2\) theorem; that is, sharp dependence of the Sobolev norm of T on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative \(A_p\) estimates which are best known, and sharp in the range \(\max \{p,p'\}\ge 3\) for the fully cancellative case.

The continuous wavelet projections algorithm: A practical spectral-feature-mining approach for crop detection

Spectroscopy can be used for detecting crop characteristics. A goal of crop spectrum analysis is to extract effective features from spectral data for establishing a detection model. An ideal spectral feature set should have high sensitivity to target parameters but low information redundancy among features. However, feature-selection methods that satisfy both requirements are lacking. To address this issue, in this study, a novel method, the continuous wavelet projections algorithm (CWPA), was developed, which has advantages of both continuous wavelet analysis (CWA) and the successive projections algorithm (SPA) for generating optimal spectral feature set for crop detection. Three datasets collected for crop stress detection and retrieval of biochemical properties were used to validate the CWPA under both classification and regression scenarios. The CWPA generated a feature set with fewer features yet achieving accuracy comparable to or even higher than those of CWA and SPA. With only two to three features identified by CWPA, an overall accuracy of 98% in classifying tea plant stresses was achieved, and high coefficients of determination were obtained in retrieving corn leaf chlorophyll content (R2 = 0.8521) and equivalent water thickness (R2 = 0.9508). The mechanism of the CWPA ensures that the novel algorithm discovers the most sensitive features while retaining complementarity among features. Its ability to reduce the data dimension suggests its potential for crop monitoring and phenotyping with hyperspectral data.

A Bounded Maximal Function Operator and Its Acting on <img src=image/13425593_01.gif> Functions

With a novel generation operator known as Spherical Scaling Wavelet Projection Operator, this study proposes new strategies for achieving the scaling wavelet expansion's convergence of <img src=image/13425593_01.gif> functions with respect to <img src=image/13425593_02.gif> almost everywhere under generic hypotheses. Hypotheses of results are based on three types of conditions: Space's function f, Kind of Wavelet functions (spherical) and Wavelet Conditions. The results showed that in the case of <img src=image/13425593_03.gif> and under the assumption that scaling wavelet function <img src=image/13425593_04.gif> of a given multiresolution analysis is spherical wavelet with 0-regularity, the convergence of <img src=image/13425593_01.gif>) expansions almost everywhere will be achieved under a new kind of partial sums operator. We can examine some properties of spherical scaling wavelet functions like rapidity of decreasing and boundedness. After estimating the bounds of spherical scaling wavelet expansions, we examined the limited (bounds) of this operator. The results are established on the almost everywhere wavelet expansions convergence of <img src=image/13425593_01.gif> space functions. Several techniques were followed to achieve this convergence, such as the bounded condition of the Spherical Hardy-Littlewood maximal operator is achieved using the maximal inequality and Riesz basis functions conditions. The general wavelet expansions' convergence was demonstrated using the spherical scaling wavelet function and several of its fundamental features. In fact, the partial sums in these expansions are dominated in their magnitude by the maximal function operator, which may be applied to establish convergence. The convergence here may be obtained by assuming minimal regularity for a spherical scaling wavelet function <img src=image/13425593_05.gif>. The focus of this research is on recent advances in convergence theory issues with respect to spherical wavelet expansions' partial sums operators. The employment of scaling wavelet basis functions defined on <img src=image/13425593_06.gif> is regarded to be a key in solving convergence problems that occur inside spaces dimension <img src=image/13425593_06.gif>.

Wavelet regularization strategy for the fractional inverse diffusion problem

This manuscript deals with an inverse fractional-diffusing problem, the time-fractional heat conduction equation, which is a physical model of a problem, where one needs to identify the temperature distribution of a semi-conductor, but one transient temperature data is unreachable to measurement. Mathematically, it is designed as a time-fractional diffusion problem in a semi-infinite region, with polluted data measured at x = 1, where the solution is wanted for 0 ≤ x < 1. In view of Hadamard, the problem extremely suffers from an intrinsic ill-posedness, i.e., the true solution of the problem is computationally impossible to measure since any measurement or numerical computation is polluted by inevitable errors. In order to capture the solution, a regularization scheme based on the Meyer wavelet is therefore applied to treat the underlying problem in the presence of polluted data. The regularized solution is restored by the Meyer wavelet projection on elements of the Meyer multiresolution analysis (MRA). Furthermore, the concepts of convergence rate and stability of the proposed scheme are investigated and some new order-optimal stable estimates of the so-called Holder-Logarithmic type are rigorously derived by carrying out both an a priori and a posteriori choice approaches in Sobolev scales. It turns out that both approaches yield the same convergence rate, except for some different constants. Finally, the computational performance of the proposed method effectively verifies the applicability and validity of our strategy. Meanwhile, the thrust of the present paper is compared with other sophisticated methods in the literature.

Nonparametric Estimation for Hazard Rate Function by Wavelet Procedures with Simulation

This article is a study of a non-parametric estimation of the hazard rate function using the linear wavelet estimation for right randomly censoring data. The strategy of the estimation is based on the use of the wavelet projection of the father function{φj,k(x),j = 1,2,…, 0 ≤ k ≤2j−1} on the subspace (Vj) of the spaceL2(R), with the Breslow estimate of the cumulative function. Real data of patients suffering from liver metastases is using as an application. Moreover, a simulation study is used to give more clarify to the method of estimation.

Multiresolution Virtual Experiments for Microwave Imaging of Complex Scenarios

In this paper, a multiresolution approach for the quantitative microwave imaging of complex scenarios is introduced. The proposed strategy takes advantage of the combined use of a recently introduced iterative method known as distorted iterated virtual experiments (DIVE), based on the paradigm of “virtual experiments”, and a wavelet-based projection scheme. This strategy allows the unknown profiles to be represented at different resolution scales and, as such, it is particularly suitable for the imaging of highly heterogeneous targets. Moreover, the developed algorithm blends together the intrinsic multiresolution feature of the wavelet projection with the one gained by means of a frequency hopping technique. The method was tested against realistic heterogeneous scenarios of practical interest, such as breast and tree trunk phantoms, which are of interest in non-invasive medical diagnostics and the health monitoring of standing trees.

Strong Lp convergence of wavelet deconvolution density estimators

Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab. 37(4) (2009) 1605–1646] obtain the optimal strong [Formula: see text] convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli 16(4) (2010) 1137–1163]. This paper considers the strong [Formula: see text] convergence of wavelet deconvolution density estimators. We first show the strong [Formula: see text] consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong [Formula: see text] convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and [Formula: see text], our convergence rate is close to Giné and Nickl’s.

Interpolatory estimates, Riesz transforms and wavelet projections

We prove that directional wavelet projections and Riesz transforms are related by interpolatory estimates. The exponents of interpolation depend on the Hölder estimates of the wavelet system. This paper complements and continues previous work on Haar projections.

A wavelet method for numerical fractional derivative with noisy data

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

An Improved Singularity Analysis for ASCAT Wind Quality Control: Application to Low Winds

Singularity analysis has proven to be a complementary tool to the Advanced Scatterometer (ASCAT) inversion residual (or maximum likelihood estimator) in terms of wind quality control (QC). In this paper, a new implementation scheme of singularity exponent (SE) is developed for ASCAT data analysis. It combines the wavelet projections of the gradient measurements of multiple parameters into the analysis, ensuring that the analyzed parameters contribute equally to the final singularity map. Therefore, the underlying geophysical phenomena in the different ASCAT-derived parameters can be effectively revealed simultaneously on a unique map of SEs. The validation using both buoy winds and European Centre for Medium-Range Weather Forecasting forecast wind output shows that the newly derived SE significantly improves the current ASCAT wind QC. In particular, poor-quality ASCAT measurements at low-wind and high-variability conditions (w <; 4 m/s) can be effectively screened using the new SE.

Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift

As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) $L^{\infty}$-diameter by using wavelet projection estimators. In addition our setting applies to the drift estimation of diffusions observed discretely with fixed observation distance. We prove a functional central limit theorem for estimators of the drift function and finally construct adaptive confidence bands for the drift by using a completely data-driven estimator.

Wavelet projection methods for solving pseudodifferential inverse problems

We consider the Inverse Problem (IP) associated to an equation of the form Af = g, where A is a pseudodifferential operator with symbol[Formula: see text]. It consists in finding a solution f for given data g. When the operator A is not strongly invertible and the data is perturbed with noise, the IP may be ill-posed and the solution must be approximate carefully. For the present application we regard a particular orthonormal wavelet basis and perform a wavelet projection method to construct a solution to the Forward Problem (FP). The approximate solution to the IP is achieved based on the decomposition of the perturbed data calculating the elementary solutions that are nearly the preimages of the wavelets. Based on properties of both, the basis and the operator, and taking into account the energy of the data, we can handle the error that arises from the partial knowledge of the data and from the non-exact inversion of each element of the wavelet basis. We estimate the error of the approximation and discuss the advantages of the proposed scheme.

On wavelet projection kernels and the integrated squared error in density estimation

Abstract It is shown that the integrated squared errors of wavelet projection estimators of a density f satisfy both the central limit theorem and the law of the iterated logarithm under the essentially minimal assumption f ∈ L p for some p > 2 and very mild conditions on the scaling function.

Average Derivative Estimation from Biased Data

We investigate the estimation of the density-weighted average derivative from biased data. An estimator integrating a plug-in approach and wavelet projections is constructed. We prove that it attains the parametric rate of convergence 1/n under the mean squared error.

A Model Predictive Control Strategy Based on Consistency in Wavelet Projections

This paper proposes a novel method of system modelling and control based on the idea of consistency in wavelet projections. Although the idea of consistency has been a familiar knowledge in the theory of generalized sampling, extension of it in the arena of system modelling and control is relatively new. It is proposed here that nonlinear approximation could be used for predicting system output that is consistent with the signal component of the desired output measurement. This imposes a number of equality and inequality constraints on wavelet projections of the input and the output. As a natural consequence, the control design problem can be viewed in the framework of multi-scale model predictive control. Idea of consistency can be further used to design a linear control law that minimizes error in least squares sense in the chosen forecast horizon. The technique of controller realization is demonstrated based on the simulation data with the point kinetics model of a nuclear reactor.